Advanced and retarded Green operators

Definition

Let $P:\Gamma^\infty(E)\to\Gamma^\infty(E)$ be a differential operator on a time-oriented Lorentzian manifold. A future Green operator $G^+$ and a past Green operator $G^-$ are linear maps

G^\pm:\Gamma^\infty_c(E)\longrightarrow\Gamma^\infty(E)

such that, for every compactly supported smooth section $f$ ,

P G^\pm f=f,\qquad G^\pm P f=f,

and

\operatorname{supp}(G^\pm f)\subseteq J^\pm(\operatorname{supp}f).

Here $J^\pm$ are the causal future and past.

These are often called the advanced and retarded Green operators. Naming conventions for “advanced” and “retarded” vary, so the signs and causal support conditions above fix the meaning unambiguously in this collection.

The Green-operator existence theorem supplies a unique pair for every normally hyperbolic operator on a globally hyperbolic spacetime.

References

Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, European Mathematical Society, 2007. Publisher record. Relevant: Definition 3.4.1.